Abstract

Inspired by Lorenz’ remarkable chaotic flow, we describe in this paper the structure of all $C^1$ robust transitive sets with singularities for flows on closed $3$-manifolds: they are partially hyperbolic with volume-expanding central direction, and are either attractors or repellers. In particular, any $C^1$ robust attractor with singularities for flows on closed $3$-manifolds always has an invariant foliation whose leaves are forward contracted by the flow, and has positive Lyapunov exponent at every orbit, showing that any $C^1$ robust attractor resembles a geometric Lorenz attractor.